Operations Research

Operations Research Session II – Sequencing

Sequencing • Scenario – • To determine the order or sequence in which the jobs are to be processed through machines so as to minimize the total processing time • A general sequencing problem may be defined as follows: • Let there be ‘n’ jobs (J1, J2, J3 ………Jn) which are to be processed on ‘m’ machines (A, B, C, ………), where the order of processing on machines i.e. for example, ABC means first on machine A, second on machine B and third on machine C or CBA means first on machine C, second on machine B and third on machine A etc. and the processing time of jobs on machines (actual or expected) is known to us, then our job is to find the optimal sequence of processing jobs that minimizes the total processing time or cost.

LPP – Problem • Any linear programming model (problem) must have the following properties: • (a) The relationship between variables and constraints must be linear. • (b) The model must have an objective function. • (c) The model must have structural constraints. • (d) The model must have non-negativity constraint.

Assumption • (a) The processing times Ai and Bi etc. are exactly known to us and they are independent of order of processing the job on the machine. That is whether job is done first on the machine, last on the machine, the time taken to process the job will not vary it remains constant. • (b) The time taken by the job from one machine to other after processing on the previous machine is negligible. (Or we assume that the processing time given also includes the transfer time and setup time). • (c) Each job once started on the machine, we should not stop the processing in the middle. It is to be processed completely before loading the next job. • (d) The job starts on the machine as soon as the job and the machine both become idle (vacant). • This is written as job is next to the machine and the machine is next to the job. (This is • exactly the meaning of transfer time is negligible). • (e) No machine may process more than one job simultaneously. (This means to say that the job once started on a machine, it should be done until completion of the processing on that • machine). • (f) The cost of keeping the semi-finished job in inventory when next machine on which the job is to be processed is busy is assumed to be same for all jobs or it is assumed that it is too small and is negligible. That is in process inventory cost is negligible. • (g) While processing, no job is given priority i.e. the order of completion of jobs has no significance. The processing times are independent of sequence of jobs. • (h) There is only one machine of each type.

Method • ‘n’ jobs are to be processed on two machines A and B in the order AB ( i.e. each job is to be processed first on A and then on B) and passing is not allowed. That is which ever job is processed first on machine A is to be first processed on machine B also, Which ever job is processed second on machine A is to be processed second on machine B also and so on. That means each job will first go to machine A get processed and then go to machine B and get processed. This rule is known as no passing rule. • Johnson and Bellman method concentrates on minimizing the idle time of machines. Johnson and Bellman have proved that optimal sequence of ‘n’ jobs which are to be processed on two machines A and B in the order AB necessarily involves the same ordering of jobs on each machine. This result also holds for three machines but does not necessarily hold for more than three machines. Thus total elapsed time is minimum when the sequence of jobs is same for both the machines

Method • Let the number of jobs be 1,2,3,…………n • The processing time of jobs on machine A be A1, A2, A3 …………. An • The processing time of jobs on machine B be B1, B2, B3 …………..Bn

Method • If there are ‘n’ jobs, first write ‘n’ number of rectangles as shown. When ever the smallest elements falls in column 1 then enter the job number in first rectangle. If it falls in second column, then write the job number in the last rectangle. Once the job number is entered, the second rectangle will become first rectangle and last but one rectangle will be the last rectangle

Method • Now calculate the total elapsed time as discussed. Write the table as shown. Let us assume that the first job starts at Zero th time. Then add the processing time of job (first in the optimal sequence) and write in out column under machine 1. This is the time when the first job in the optimal sequence leaves machine 1 and enters the machine 2. Now add processing time of job on machine 2. This is the time by which the processing of the job on two machines over. Next consider the job, which is in second place in optimal sequence. This job enters the machine 1 as soon the machine becomes vacant, i.e first job leaves to second machine. Hence enter the time in out column for first job under machine 1 as the starting time of job two on machine 1. Continue until all the jobs are over. Be careful to see that whether the machines are vacant before loading. Total elapsed time may be worked out by drawing Gantt chart for the optimal sequence

Points to remember

Problem • Processing times in hours for the jobs are given below. Find the optimal sequence and total elapsed time. (Students has to remember in sequencing problems if optimal sequence is asked, it is the duty of the student to find the total elapsed time also).

Solution

Example

Exercise

Graph - 1

Combined

Simplex Method SIMPLEX METHOD is considered as the most powerful method of LPP . It deals with iterative process, which consists of first designing a Basic Feasible Solution or a Program and proceed towards the OPTIMAL SOLUTION and testing each feasible solution for Optimality to know whether the solution on hand is optimal or not. If not an optimal solution, redesign the program, and test for optimality until the test confirms OPTIMALITY. We thus can say that the Simplex Method depends on two concepts known as Feasibility and optimality

LPP assumptions • Certainty • Linearity • Divisibility • Single stage • Non-negativity

Transportation • The transportation model deals with a special class of linear programming problem in which the objective is to transport a homogeneous commodity from various origins or factories to different destinations or markets at a total minimum cost. • Example

Solution to Transportation • North West Corner Method • Least Cost Method • Vogel’s Approximation Method

Operations Research